The exponent of a primitive digraph is the smallest integer k such that for each ordered pair of (not necessarily distinct) vertices x and y there is a walk of length k from x to y. As a generalization of exponent, Brualdi and Liu (Linear Algebra Appl. 14 (1990) 483499) introduced three types of generalized exponents for primitive digraphs in 1990. In this Paper we extend their definitions of g...

In the Directed k-Chinese Postman Problem (k-DCPP), we are given a connected weighted digraph G and asked to find k non-empty closed directed walks covering all arcs of G such that the total weight of the walks is minimum. Gutin, Muciaccia and Yeo (Theor. Comput. Sci. 513 (2013) 124–128) asked for the parameterized complexity of k-DCPP when k is the parameter. We prove that the k-DCPP is fixed-...

If D = (V,A) is a digraph, its competition hypergraph CH(D) has vertex set V and e ⊆ V is an edge of CH(D) iff |e| ≥ 2 and there is a vertex v ∈ V , such that e = N− D (v) = {w ∈ V |(w, v) ∈ A}. We give characterizations of CH(D) in case of hamiltonian digraphs D and, more general, of digraphs D having a τ -cycle factor. The results are closely related to the corresponding investigations for co...

A (k1 + k2)-bispindle is the union of k1 (x, y)-dipaths and k2 (y, x)-dipaths, all these dipaths being pairwise internally disjoint. Recently, Cohen et al. showed that for every (1, 1)bispindle B, there exists an integer k such that every strongly connected digraph with chromatic number greater than k contains a subdivision of B. We investigate generalisations of this result by first showing co...

Local route discovery in geometric, connected, undirected plane graphs is guaranteed by the Face Routing algorithm. The algorithm is local and geometric in the sense that it is executed by an agent moving along a network and using at each node only information about the current node (incl. its position) and a finite number of others (independent of graph size). Local geometric traversal algorit...

An out-(in-)branching B s (B − s ) rooted at s in a digraph D is a connected spanning subdigraph of D in which every vertex x 6= s has precisely one arc entering (leaving) it and s has no arcs entering (leaving) it. We settle the complexity of the following two problems: • Given a 2-regular digraph D, decide if it contains two arc-disjoint branchings B u , B − v . • Given a 2-regular digraph D,...